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arxiv: 2301.07055 · v3 · pith:IBP5DXRAnew · submitted 2023-01-17 · 🧮 math.AG

On G-birational rigidity of del Pezzo surfaces

Egor Yasinsky This is my paper

Pith reviewed 2026-05-24 09:51 UTC · model grok-4.3

classification 🧮 math.AG
keywords del Pezzo surfacesbirational rigidityfinite group actionsbirational geometryalgebraic surfaces
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The pith

If a smooth del Pezzo surface is birationally rigid under a subgroup of a finite group, then it is also rigid under the full group.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that H-birational rigidity implies G-birational rigidity for any finite groups H contained in G acting on a smooth del Pezzo surface over an algebraically closed field. This transfers a property verified for the smaller group directly to the larger one without additional checks. A reader would care because it reduces the work of establishing rigidity when multiple group actions are possible on the same surface. The result gives a positive answer in dimension two to a geometric form of a question about how rigidity behaves under group enlargement.

Core claim

The paper proves that if a smooth del Pezzo surface over an algebraically closed field is H-birationally rigid for a subgroup H of a finite group G, then the surface is also G-birationally rigid.

What carries the argument

The direct implication that H-rigidity forces G-rigidity for finite group actions on the surface.

If this is right

  • Rigidity statements for del Pezzo surfaces need only be checked against minimal subgroups rather than every possible finite group.
  • Any known H-rigid example automatically yields a G-rigid example for every supergroup G containing H.
  • The set of finite groups under which a given surface is birationally rigid is closed upward under inclusion.
  • Classification efforts for birationally rigid del Pezzo surfaces can focus on maximal or minimal group actions without loss of generality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same upward-closure property might hold for other classes of surfaces or varieties once the dimension-two case is settled.
  • If the implication fails in higher dimensions, the failure would have to arise from phenomena absent on del Pezzo surfaces.
  • Concrete lists of rigid del Pezzo surfaces under small groups can now be enlarged mechanically to their supergroups.

Load-bearing premise

The del Pezzo surface must be smooth and the base field algebraically closed, with G finite and H a subgroup of G.

What would settle it

An explicit smooth del Pezzo surface together with finite groups H properly contained in G such that the surface is H-birationally rigid but fails to be G-birationally rigid.

Figures

Figures reproduced from arXiv: 2301.07055 by Egor Yasinsky.

Figure 1
Figure 1. Figure 1: Action of 𝑠 on Σ Lemma 3.5. Let 𝑆 be a del Pezzo surface of degree 6 and 𝐺 ⊂ Aut(𝑆) be a finite group such that Pic(𝑆) 𝐺 ≃ Z. If 𝐺 fixes a point on 𝑆 then 𝐺 ∩ 𝑇 = id. Proof. Assume 𝐺 ∩ 𝑇 ̸= id. Note that 𝑇 can be identified with a subgroup of PGL3(k) which fixes 3 points 𝑝1, 𝑝2 and 𝑝3. In particular, an element 𝑡 ∈ 𝑇 fixing a point on 𝑆 ∖ Σ is necessarily trivial (of course, one can also deduce that from t… view at source ↗
Figure 2
Figure 2. Figure 2: The Clebsch graph Now, one easily checks that involutions 𝜄*5 = 𝚥* ∘ 𝚥5 and 𝜄*4 = 𝚥* ∘ 𝚥4 do not preserve the set Σ. Hence, the subgroup of W(𝐷5) which preserves Σ is generated by (12),(123), 𝜄12 and (45), and is isomorphic to S4 × C2. The involution 𝜄45 commutes with this group and maps Σ onto Σ′ . Since 𝜄45 actually corresponds to an automorphism of 𝑇, we conclude that the blow down 𝜂 ′ yields a 𝐺-isomor… view at source ↗
read the original abstract

Let $G$ be a finite group and $H\subseteq G$ be its subgroup. We prove that if a smooth del Pezzo surface over an algebraically closed field is $H$-birationally rigid then it is also $G$-birationally rigid, answering a geometric version of Koll\'{a}r's question in dimension 2 by positive.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript proves that if a smooth del Pezzo surface over an algebraically closed field is H-birationally rigid, where H is a subgroup of a finite group G, then it is also G-birationally rigid. This is presented as a positive answer to a geometric version of Kollár's question in dimension 2.

Significance. If the result holds, it resolves the stated question positively in dimension 2. The argument follows directly from the definitions of equivariant birational maps (any G-equivariant map restricts to an H-equivariant map, so H-rigidity forces it to be an isomorphism that remains G-equivariant), without invoking special properties of del Pezzo surfaces, dimension 2, or algebraic closure. This generality strengthens the logical step but limits the geometric novelty.

minor comments (1)
  1. The abstract states the result but provides no outline of the argument; expanding it slightly would improve accessibility without altering the content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive recommendation to accept the manuscript. The referee correctly notes that the core implication follows directly from the definitions of equivariant birational maps.

Circularity Check

0 steps flagged

No circularity; direct implication from definitions of rigidity

full rationale

The paper proves that H-birational rigidity implies G-birational rigidity for smooth del Pezzo surfaces when H ≤ G. This follows immediately once the definitions are fixed: a G-equivariant birational map to another G-variety is automatically H-equivariant (by restriction of the group action), so the H-rigidity hypothesis forces it to be an isomorphism, which is then G-equivariant. No equations, fitted parameters, self-citations, or ansatzes are invoked in the load-bearing step; the result is a pure logical consequence of the setup. The paper is self-contained as a proof of this implication with no reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard background from algebraic geometry rather than new parameters or entities. No free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Smooth del Pezzo surfaces over algebraically closed fields and finite group actions are defined and behave according to standard birational geometry.
    This is the setup assumed for the statement.
  • domain assumption The notion of H-birational rigidity is the established one in the literature.
    The paper uses the term without redefinition.

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Reference graph

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32 extracted references · 32 canonical work pages · 1 internal anchor

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